Beat eggs together in a large bowl. Gradually stir white sugar into beaten eggs until smooth. Slowly pour vegetable oil and anise oil into sugar mixture until incorporated. Mix flour and baking powder together in a separate bowl; slowly add to sugar mixture, stirring with a wooden spoon until dough is dry.
Refrigerate dough, 30 minutes to overnight.
Preheat oven to 350 degrees F (175 degrees C). Lightly grease a baking sheet.
Roll dough into walnut-size balls and arrange on the prepared baking sheet.
Bake in the preheated oven until cookies are crisp around the edges, 12 to 15 minutes. Cool cookies on baking sheet for 5 minutes before transferring to a wire rack.
Mix confectioners' sugar and milk together in a bowl until desired consistency is reached. Dip a fork into the icing and drizzle over cookies. Allow icing to harden.